| L(s) = 1 | − 2·3-s + 4·5-s + 3·9-s + 4·11-s + 4·13-s − 8·15-s + 4·17-s − 8·23-s + 4·25-s − 4·27-s + 12·29-s − 8·33-s + 12·37-s − 8·39-s − 4·41-s − 8·43-s + 12·45-s − 8·47-s − 12·49-s − 8·51-s + 12·53-s + 16·55-s + 8·59-s + 12·61-s + 16·65-s + 16·69-s − 8·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 9-s + 1.20·11-s + 1.10·13-s − 2.06·15-s + 0.970·17-s − 1.66·23-s + 4/5·25-s − 0.769·27-s + 2.22·29-s − 1.39·33-s + 1.97·37-s − 1.28·39-s − 0.624·41-s − 1.21·43-s + 1.78·45-s − 1.16·47-s − 1.71·49-s − 1.12·51-s + 1.64·53-s + 2.15·55-s + 1.04·59-s + 1.53·61-s + 1.98·65-s + 1.92·69-s − 0.923·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.032859282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.032859282\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 32 T + 412 T^{2} - 32 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.747655769889656398421570804949, −9.692040080699456201586394716886, −8.788142312365327811825931283203, −8.605504592467601746638323401608, −7.920490330195012282970678709784, −7.889123220161755358339351386077, −6.83841876415996400474826198960, −6.60546057390317598997669417926, −6.33631621673338647429093132414, −6.10540565589525099056978718055, −5.58627091357010220119804614420, −5.36263466008609156728625107636, −4.68214789988357843469181011262, −4.37488978892752516694136497168, −3.54705160951407361948314721116, −3.44369405903479922806992837690, −2.27570647898900023132084560997, −2.00506838911489618005968151923, −1.23702908956003681583750718601, −0.866555615128013867607089515377,
0.866555615128013867607089515377, 1.23702908956003681583750718601, 2.00506838911489618005968151923, 2.27570647898900023132084560997, 3.44369405903479922806992837690, 3.54705160951407361948314721116, 4.37488978892752516694136497168, 4.68214789988357843469181011262, 5.36263466008609156728625107636, 5.58627091357010220119804614420, 6.10540565589525099056978718055, 6.33631621673338647429093132414, 6.60546057390317598997669417926, 6.83841876415996400474826198960, 7.889123220161755358339351386077, 7.920490330195012282970678709784, 8.605504592467601746638323401608, 8.788142312365327811825931283203, 9.692040080699456201586394716886, 9.747655769889656398421570804949
